Recall that a knot is amphichiral (or achiral) if there is a continuous deformation of the knot into its mirror image.
I'm interested in knowing when and whether we can use approaches like Stockmeyer counting to estimate and distinguish different knots. For example suppose that for most knots of grid dimension $d$, the total number of grid diagrams isotopic to a given knot is $\mathcal O(n)$. Then given two knot diagrams/grid diagrams $K_1$ and $K_2$ we can perform random Reidemeister moves on either $K_1$ or $K_2$ (allowing for a small increase in the grid dimension if needed), and hash the grid diagrams to do the counting. If $K_1$ and $K_2$ represent two distinct knots then there might be double $\mathcal O(2n)$ the number of such grid diagrams than if $K_1$ and $K_2$ represent the same knot.
Nonetheless if there's a sequence of such moves that transforms $K_1$ into a mirror image $K_2$ then the provided knot is amphichiral. But do we have any intuition on whether this would then mean that Stockmeyer counting would still say that the total number of grid diagrams of $K_1$ and $K_2$ together are twice as much as the total number of grid diagrams of another random knot $K_0$ of the same grid number $d$?
The Wikipedia article gives representative OEIS ID's for the number of amphichiral knots, and amphichirality appears rare.